K-stability of smooth Fano hypersurfaces of intermediate degrees

Determine whether smooth Fano hypersurfaces of intermediate degrees are K-stable.

Background

The introduction surveys known K-stability results for smooth Fano hypersurfaces. Smooth Fano hypersurfaces of Fano index one and two are known to be K-stable, and a result of Abban and Zhuang establishes K-stability for Fano index at least three when the dimension is sufficiently large. The authors identify the remaining intermediate-degree cases as unresolved.

References

The K-stability problem for smooth hypersurfaces of intermediate degrees remains open.

— K-stability of a class of smooth Fano complete intersections  (2610.03311 - Hu, 2 Oct 2026) in Section 1, Introduction